3.2.27 \(\int \frac {a+b \arcsin (c x)}{x^3 (d-c^2 d x^2)^{3/2}} \, dx\) [127]

3.2.27.1 Optimal result

Integrand size = 27, antiderivative size = 316 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b c^2 \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 d \sqrt {d-c^2 d x^2}} \]

output

3/2*c^2*(a+b*arcsin(c*x))/d/(-c^2*d*x^2+d)^(1/2)+1/2*(-a-b*arcsin(c*x))/d/ 
x^2/(-c^2*d*x^2+d)^(1/2)-1/2*b*c*(-c^2*x^2+1)^(1/2)/d/x/(-c^2*d*x^2+d)^(1/ 
2)-3*c^2*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^ 
(1/2)/d/(-c^2*d*x^2+d)^(1/2)-b*c^2*arctanh(c*x)*(-c^2*x^2+1)^(1/2)/d/(-c^2 
*d*x^2+d)^(1/2)+3/2*I*b*c^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2 
+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-3/2*I*b*c^2*polylog(2,I*c*x+(-c^2*x^2+1)^ 
(1/2))*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)
 

3.2.27.2 Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\frac {4 a \sqrt {d} \left (-1+3 c^2 x^2\right )}{x^2 \sqrt {d-c^2 d x^2}}+12 a c^2 \log (x)-12 a c^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b \sqrt {d} \left (2 \arcsin (c x)-6 \arcsin (c x) \cos (2 \arcsin (c x))-3 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )+3 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )-2 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+\sqrt {1-c^2 x^2} \left (3 \arcsin (c x) \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )+2 \left (\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )\right )+2 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-2 \sin (2 \arcsin (c x))+6 i c x \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-6 i c x \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))\right )}{x^2 \sqrt {d-c^2 d x^2}}}{8 d^{3/2}} \]

input

Integrate[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(3/2)),x]
 

output

((4*a*Sqrt[d]*(-1 + 3*c^2*x^2))/(x^2*Sqrt[d - c^2*d*x^2]) + 12*a*c^2*Log[x 
] - 12*a*c^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*Sqrt[d]*(2*ArcSin[c 
*x] - 6*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]]* 
Log[1 - E^(I*ArcSin[c*x])] + 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I 
*ArcSin[c*x])] - 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[ 
c*x]/2]] + Sqrt[1 - c^2*x^2]*(3*ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - 
Log[1 + E^(I*ArcSin[c*x])]) + 2*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/ 
2]] - Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])) + 2*Cos[3*ArcSin[c*x] 
]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 2*Sin[2*ArcSin[c*x]] + (6 
*I)*c*x*PolyLog[2, -E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - (6*I)*c*x*Poly 
Log[2, E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2]))/ 
(8*d^(3/2))
 

3.2.27.3 Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.79, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5204, 264, 219, 5208, 219, 5218, 3042, 4671, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 5204

\(\displaystyle \frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (c^2 \int \frac {1}{1-c^2 x^2}dx-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5208

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5218

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

input

Int[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(3/2)),x]
 

output

-1/2*(a + b*ArcSin[c*x])/(d*x^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x 
^2]*(-x^(-1) + c*ArcTanh[c*x]))/(2*d*Sqrt[d - c^2*d*x^2]) + (3*c^2*((a + b 
*ArcSin[c*x])/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x]) 
/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(-2*(a + b*ArcSin[c*x])*ArcT 
anh[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[ 
2, E^(I*ArcSin[c*x])]))/(d*Sqrt[d - c^2*d*x^2])))/2
 

3.2.27.3.1 Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5204

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* 
c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*( 
1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 5208

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c 
*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)* 
(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b 
, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

rule 5218

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* 
x^2]]   Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a 
, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

3.2.27.4 Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.34

method	result	size
default	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{2 d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\)	\(425\)
parts	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{2 d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\)	\(425\)

input

int((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

output

a*(-1/2/d/x^2/(-c^2*d*x^2+d)^(1/2)+3/2*c^2*(1/d/(-c^2*d*x^2+d)^(1/2)-1/d^( 
3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)))-1/2*I*b*(-d*(c^2*x^2-1)) 
^(1/2)*(-c^2*x^2+1)^(1/2)*(3*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))* 
x^4*c^4+3*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+4*arctan(I*c*x+(-c^2*x^2 
+1)^(1/2))*c^4*x^4+3*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+3*I*(-c^2*x 
^2+1)^(1/2)*arcsin(c*x)*c^2*x^2-3*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1 
/2))*x^2*c^2+I*x^3*c^3-3*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-4*arctan( 
I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-3*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^2* 
x^2-I*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-I*c*x)/d^2/(c^4*x^4-2*c^2*x^2+1)/x^2
 

3.2.27.5 Fricas [F]

\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

input

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas" 
)
 

output

integral(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^4*d^2*x^7 - 2*c^2*d^2 
*x^5 + d^2*x^3), x)
 

3.2.27.6 Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

input

integrate((a+b*asin(c*x))/x**3/(-c**2*d*x**2+d)**(3/2),x)
 

output

Integral((a + b*asin(c*x))/(x**3*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)
 

3.2.27.7 Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

input

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima" 
)
 

output

-1/2*(3*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(3/2 
) - 3*c^2/(sqrt(-c^2*d*x^2 + d)*d) + 1/(sqrt(-c^2*d*x^2 + d)*d*x^2))*a - b 
*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^2*d*x^5 - d*x^3) 
*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
 

3.2.27.8 Giac [F]

\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

input

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")
 

output

integrate((b*arcsin(c*x) + a)/((-c^2*d*x^2 + d)^(3/2)*x^3), x)
 

3.2.27.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

input

int((a + b*asin(c*x))/(x^3*(d - c^2*d*x^2)^(3/2)),x)
 

output

int((a + b*asin(c*x))/(x^3*(d - c^2*d*x^2)^(3/2)), x)